WITHDRAWN: A lower bound for blow-up in a model for aggregation of microglia

2012 ◽  
Author(s):  
J.C. Song
Keyword(s):  
Lower Bound ◽  
Blow Up

scholarly journals A lower bound for blow-up time in a fully parabolic Keller–Segel system

2013 ◽  
Vol 26 (4) ◽  
pp. 510-514 ◽  
Author(s):  
Jingru Li ◽  
Sining Zheng
Keyword(s):  
Lower Bound ◽  
Blow Up

Lower bound for the perimeter density at singular points of a minimizing cluster in ℝN

2020 ◽  
Vol 26 ◽  
pp. 1
Author(s):  
Jonas Hirsch ◽  
Michele Marini
Keyword(s):  
Lower Bound ◽  
Blow Up ◽  
Singular Points ◽  
Sharp Lower Bound

In this paper, we study the blow-ups of the singular points in the boundary of a minimizing cluster lying in the interface of more than two chambers. We establish a sharp lower bound for the perimeter density at those points and we prove that this bound is rigid, namely having the lowest possible density completely characterizes the blow-up.

scholarly journals BÜCHI COMPLEMENTATION MADE TIGHTER

2006 ◽  
Vol 17 (04) ◽  
pp. 851-867 ◽  
Author(s):  
EHUD FRIEDGUT ◽  
ORNA KUPFERMAN ◽  
MOSHE Y. VARDI
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Blow Up ◽  
Careful Analysis ◽  
Point Of View ◽  
Upper And Lower Bounds ◽  
Theoretical Point ◽  
Infinite Words ◽  
Büchi Automata ◽  
Buchi Automata

The complementation problem for nondeterministic word automata has numerous applications in formal verification. In particular, the language-containment problem, to which many verification problems is reduced, involves complementation. For automata on finite words, which correspond to safety properties, complementation involves determinization. The 2n blow-up that is caused by the subset construction is justified by a tight lower bound. For Büchi automata on infinite words, which are required for the modeling of liveness properties, optimal complementation constructions are quite complicated, as the subset construction is not sufficient. From a theoretical point of view, the problem is considered solved since 1988, when Safra came up with a determinization construction for Büchi automata, leading to a 2O(n log n) complementation construction, and Michel came up with a matching lower bound. A careful analysis, however, of the exact blow-up in Safra's and Michel's bounds reveals an exponential gap in the constants hiding in the O( ) notations: while the upper bound on the number of states in Safra's complementary automaton is n2n, Michel's lower bound involves only an n! blow up, which is roughly (n/e)n. The exponential gap exists also in more recent complementation constructions. In particular, the upper bound on the number of states in the complementation construction of Kupferman and Vardi, which avoids determinization, is (6n)n. This is in contrast with the case of automata on finite words, where the upper and lower bounds coincides. In this work we describe an improved complementation construction for nondeterministic Büchi automata and analyze its complexity. We show that the new construction results in an automaton with at most (0.96n)n states. While this leaves the problem about the exact blow up open, the gap is now exponentially smaller. From a practical point of view, our solution enjoys the simplicity of the construction of Kupferman and Vardi, and results in much smaller automata.

scholarly journals On a system of nonlinear pseudoparabolic equations with Robin-Dirichlet boundary conditions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Le Thi Phuong Ngoc ◽  
Khong Thi Thao Uyen ◽  
Nguyen Huu Nhan ◽  
Nguyen Thanh Long
Keyword(s):  
Weak Solution ◽  
Lower Bound ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Local Existence ◽  
Initial Energy ◽  
Dirichlet Boundary Conditions ◽  
Local Existence And Uniqueness ◽  
Pseudoparabolic Equations

<p style='text-indent:20px;'>In this paper, we investigate a system of pseudoparabolic equations with Robin-Dirichlet conditions. First, the local existence and uniqueness of a weak solution are established by applying the Faedo-Galerkin method. Next, for suitable initial datum, we obtain the global existence and decay of weak solutions. Finally, using concavity method, we prove blow-up results for solutions when the initial energy is nonnegative or negative, then we establish here the lifespan for the equations via finding the upper bound and the lower bound for the blow-up times.</p>

Bounds for blow-up time for the heat equation under nonlinear boundary conditions

2009 ◽  
Vol 139 (6) ◽  
pp. 1289-1296 ◽  
Author(s):  
L. E. Payne ◽  
P. W. Schaefer
Keyword(s):  
Boundary Condition ◽  
Lower Bound ◽  
Heat Equation ◽  
Differential Inequality ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Blow Up ◽  
Nonlinear Boundary Conditions ◽  
Sufficient Condition ◽  
Inequality Technique

A differential inequality technique is used to determine a lower bound on the blow-up time for solutions to the heat equation subject to a nonlinear boundary condition when blow-up of the solution does occur. In addition, a sufficient condition which implies that blow-up does occur is determined.

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